Applications of Derivatives — Class 12 Maharashtra HSC Mathematics revision notes
A 5-page visual revision sheet for Applications of Derivatives, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Applications of Derivatives in one look
one slope reused six ways: a tangent line, a rate, an estimate, two named theorems, and a turning point
Tangent and Normal
- slope of the tangent at P(x_1,y_1), m = dy/dx evaluated there
m = left(dy/dxright)_(x_1,y_1) - slope of the normal at the same point, perpendicular to the tangent
m' = -1/m, quad m≠0 - equation of the tangent at P(x_1,y_1)
y-y_1 = m(x-x_1) - equation of the normal at P(x_1,y_1)
y-y_1 = m'(x-x_1)
Tangent and normal to a cubic
Q. Find the equations of the tangent and normal to y=2x^3-x^2+2 at the point (1/2, 2).
- dy/dx = 6x^2-2x
- at x=1/2: m = 6(1/4)-2(1/2) = 3/2-1 = 1/2
- tangent: y-2 = (1/2)(x-1/2), i.e. 2y-4 = x-1/2, so 2x-4y+7=0
- normal slope m' = -1/m = -2: y-2 = -2(x-1/2), i.e. y-2 = -2x+1
Answer: tangent: 2x-4y+7=0; normal: 2x+y-3=0
Special tangent cases
- a horizontal tangent happens exactly where dy/dx=0: the curve momentarily stops rising or falling
- a vertical tangent happens where dx/dy=0 instead, since dy/dx itself would be undefined there
- for an implicit curve, differentiate implicitly for dy/dx in x and y, then substitute the point
Derivative as a Rate Measure
- instantaneous rate of change of y w.r.t. x
dy/dx - when two quantities x and y both change with time t
dy/dt = dy/dxcdotdx/dt
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What this Applications of Derivatives sheet covers
- Tangent and Normal
- slope of the tangent at P(x_1,y_1), m = dy/dx evaluated there
- slope of the normal at the same point, perpendicular to the tangent
- equation of the tangent at P(x_1,y_1)
- equation of the normal at P(x_1,y_1)
- Tangent and normal to a cubic
- Special tangent cases
- Derivative as a Rate Measure
- instantaneous rate of change of y w.r.t. x
- when two quantities x and y both change with time t
- Rate measure on a circle
- Approximations
- for a differentiable f and small h, the value near x=a
- Approximate a square root
- Rolle's Theorem
- Verify Rolle's theorem
- Lagrange's Mean Value Theorem
- conditions: f continuous on [a,b], differentiable on (a,b)
- Verify LMVT
- Increasing vs decreasing, tested with f prime
- Increasing on an interval
- Decreasing on an interval
- Intervals of increase and decrease
- Maxima and Minima
- a critical (stationary) point of f: x=c where f'(c)=0
- second derivative test: negative at a critical point means a local maximum
- positive at a critical point means a local minimum
- Second derivative test: reading f''(c)
- Classify a critical point where f prime is zero
- Maxima and minima by the second derivative test
- Optimisation in context
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Applications of Derivatives the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.